Dilatations and defect computation

For an affine chart $\mathrm{Spec}A\subseteq X$ with ideal $I\subseteq A$ cutting out $Y$ on the special fibre and containing $\pi$, the $\pi$-chart of the blowup has coordinate ring $A[I\mathrm{/}\pi ]=A[t_j:j]\mathrm{/}(\pi t_j-g_j)$ modulo its $\pi$-power torsion, by [F1]. This ring is $\pi$-torsion-free by construction, hence flat over the DVR $R$ by [F3]. If $B$ is flat over $R$ and the special morphism $B_k\to X_k$ factors through $Y$, then the images of the generators $g_j$ in $B$ are divisible by $\pi$: they vanish modulo $\pi$ because the factorization makes them lie in $IB$, so $g_j=\pi h_j$ with $h_j\in B$ unique, multiplication by $\pi$ being injective on the flat, hence $\pi$-torsion-free ring $B$. Sending $t_j\mapsto h_j$ kills all torsion and defines the unique $R$-morphism from $\mathrm{Spec}B$ to the $\pi$-chart; the blowup is locally H-projective over $X$, hence proper, and the valuative criterion gives the unique lifting of sections.

For an unramified flat extension of DVRs $R\to R^{\mathrm{\prime }}$, $\pi$ remains a uniformizer up to a unit. The inclusion $A[I\mathrm{/}\pi ]\subseteq A[1\mathrm{/}\pi ]$ stays injective after tensoring with $R^{\mathrm{\prime }}$, and its image is precisely the subalgebra generated by $A{\otimes }_R{R}^{\mathrm{\prime }}$ and the images $g_j\mathrm{/}\pi$. Thus it is the dilatation algebra after base change. For two flat models, the tensor product of their dilatation algebras is flat over $R$ and is generated over $A_1{\otimes }_R{A}_2$ by the fractions from both centre ideals; the product centre has ideal $I_1(A_1{\otimes }_R{A}_2)+I_2(A_1{\otimes }_R{A}_2)$. The universal property checked componentwise therefore identifies this tensor product with the product-centre dilatation. Finally, for a flat closed subscheme with affine ring $A\mathrm{/}J$, the homomorphism $A[I\mathrm{/}\pi ]\to (A\mathrm{/}J)[\bar{I}\mathrm{/}\pi ]$, where $\bar{I}$ is the image of $I$, is surjective: the target is generated by the images of $A$ and $g_j\mathrm{/}\pi$, and $\pi$-power torsion maps to zero. These affine surjections glue to the asserted closed immersion. The unique local factorizations in step 1.1 likewise glue for any flat source scheme $B$.

Let $f:V\to W$ be a morphism between smooth $R$-schemes of equal relative dimension $d$. Near $f(v)$ choose etale coordinates $W\to {\mathbb{A}}_R^d$, using a unit minor of a standard smooth presentation, and denote the pulled-back coordinate functions on $V$ by $f_1,\dots ,f_d$. If the relative differential determinant of $f$ is a unit at $v$, the $\mathrm{d}{f}_i$ form a basis of ${\mathrm{\Omega }}_{V\mathrm{/}R}$ there. In a standard smooth presentation of $V$ in $n$ variables, append the $d$ graph equations $T_i-f_i$ to its $n-d$ relation equations. Their differentials have a unit $n\times n$ minor, so the composite $V\to {\mathbb{A}}_R^d$ is etale at $v$ by [F2]. Because $W\to {\mathbb{A}}_R^d$ is etale, this implies $f$ is etale: for a nilpotent lifting problem over $W$, unique lifting over ${\mathbb{A}}_R^d$ first gives the lift into $V$, and formal unramifiedness of $W\to {\mathbb{A}}_R^d$ forces its composite into $W$ to be the prescribed map. Local finite presentation then gives etaleness by [F2]. Conversely, if $f$ is etale, the same lifting property identifies its relative differential map with an isomorphism of the two locally free rank-$d$ modules, so its determinant is a unit.

For a section $a$ of $X$ with smooth generic fibre of dimension $d$, the module $a^{\ast }{\mathrm{\Omega }}_{X\mathrm{/}R}$ is finitely generated over the DVR, hence the direct sum of a free part and a torsion part, and $\delta (a)$ is the length of that torsion part. If $\delta (a)=0$, the differential vector space at the rational specialization of $a$ has dimension $d$; the generic section specializes to the special section, so the upper semicontinuity of local fibre dimension [F2] gives special local dimension at least $d$, while tangent dimension bounds it above by $d$. Choosing $n-d$ local equations with independent differentials exhibits a standard smooth ambient $Z$ of relative dimension $d$ containing $X$ locally along $a$; flatness makes the local dimension of $X$ equal to its special-fibre dimension plus one, namely $d+1$, which is the local dimension of the smooth ambient $Z$ at that rational point. A regular local domain has no nonzero ideal whose quotient has the same dimension, so the defining ideal of $X$ in $Z$ is zero locally. Hence $X$ agrees with the smooth ambient near the specialization and $a$ factors through the smooth locus. Hence $X$ is smooth along $a$, and conversely smoothness makes $a^{\ast }\mathrm{\Omega }$ locally free, so its torsion vanishes.

Choose a finite affine cover $\mathrm{Spec}{A}_{\alpha }$ of $X$ and presentations $A_{\alpha }=R[T_1,\dots ,T_{n_{\alpha }}]\mathrm{/}(f_1,\dots ,f_{m_{\alpha }})$. Put $q_{\alpha }=n_{\alpha }-d$. A section whose specialization lies in this chart factors through the chart, since an open subset of $\mathrm{Spec}R$ containing its closed point is the whole spectrum. Evaluation of the Jacobian gives a presentation $R^{m_{\alpha }}\to R^{n_{\alpha }}\to a^{\ast }{\mathrm{\Omega }}_{X\mathrm{/}R}\to 0$ of generic rank $q_{\alpha }$. Smith normal form over the DVR shows that the torsion length is the sum of the valuations of its $q_{\alpha }$ nonzero diagonal entries; equivalently it is the minimum valuation of the $q_{\alpha }\times q_{\alpha }$ minors. This proves the asserted numerical formula, with the size-zero minor interpreted as $1$.

Let $J_{\alpha }\subseteq A_{\alpha }$ be the ideal generated by these minors before evaluation. Smoothness of the pure relative-dimension-$d$ generic fibre says the Jacobian has rank $q_{\alpha }$ at every generic-fibre point, hence $J_{\alpha }{A}_{\alpha }[1\mathrm{/}\pi ]=A_{\alpha }[1\mathrm{/}\pi ]$. Expressing $1$ as a finite linear combination of minors over $A_{\alpha }[1\mathrm{/}\pi ]$ and clearing denominators gives ${\pi }^{N_{\alpha }}\in J_{\alpha }$ for some $N_{\alpha }\ge 0$. After evaluating any $R^{\mathrm{s}\mathrm{h}}$-section in this chart, the minor ideal therefore contains ${\pi }^{N_{\alpha }}$, so step 5.1 gives $\delta (a)\le N_{\alpha }$. A section over $R^{\mathrm{s}\mathrm{h}}$ factors through a chart containing its specialization just as above, and the finite maximum ${\max }_{\alpha }{N}_{\alpha }$ gives the uniform bound. Finally, step 4.1 identifies defect zero with smoothness along the section and with a $d$-dimensional special cotangent space; step 3.1 supplies the determinant criterion in (c).